ExampleConstruction: Literature-sourcedVerification: AI-adaptedPipeline-generated

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Homogeneous Poisson rate estimation at positive-count and zero-count boundaries

Example

Take $W = [0, 2]^{2}$ , so $∣ W ∣ = 4$ , and fit the homogeneous model $λ_{ρ} (x) = ρ$ . For an observation with three points, the likelihood is maximized at $\hat{ρ} = 3 / 4$ . For an empty observation the likelihood increases as $ρ$ decreases to zero; the boundary value is attained only when $ρ = 0$ is included in the parameter space.

Facts & Assumptions

Given: $W = [0, 2]^{2}$ , $∣ W ∣ = 4$ , homogeneous rate $ρ$ , and the two observed counts $n = 3$ and $n = 0$ .

[F1]

For $∣ W ∣ > 0$ , a positive observed count $n$ has unique MLE $n / ∣ W ∣$ ; at zero count the supremum is the rate-zero boundary and is attained exactly when zero is admitted. (Homogeneous Poisson rate MLE in a positive-volume window)

Verification

Given: $W = [0, 2]^{2}$ , $∣ W ∣ = 4$ , homogeneous rate $ρ$ , and the two observed counts $n = 3$ and $n = 0$ .

[F1] For $∣ W ∣ > 0$ , a positive observed count $n$ has unique MLE $n / ∣ W ∣$ ; at zero count the supremum is the rate-zero boundary and is attained exactly when zero is admitted. (Homogeneous Poisson rate MLE in a positive-volume window)

1.1givenF1algebra

For $n = 3$ , [F1] gives $\hat{ρ} = n / ∣ W ∣ = 3 / 4$ . The log likelihood up to a constant is $3 \log ρ - 4 ρ$ , and the proposition proves this rate is the unique maximizer.

2.1givenF1algebra∎

For $n = 0$ , that log likelihood reduces to $- 4 ρ$ for $ρ > 0$ ; it tends to $0$ as $ρ ↓ 0$ and is strictly smaller at every positive rate. The normalized likelihood at $ρ = 0$ is positive for the empty configuration, so the maximum is attained at zero if the parameter space contains zero, while on $(0, \infty)$ only the boundary supremum exists.

Depends on

Used by

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Dependency tree · two levels

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Sources

Adrian Baddeley, Spatial Point Processes and their Applications, complete lecture notes (standard reference, not scraped)
Günter Last and Mathew Penrose, Lectures on the Poisson Process, author-hosted 2017 version (standard reference, not scraped)